Graph Reconstruction
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1 Graph Reconstruction Robert D. Borgersen Graduate Seminar: Graph Reconstruction p. 1/2
2 Abstract The concept of a graph is one of the most basic and readily understood mathematical concepts, and the Reconstruction Conjecture is one of the most engaging problems under the domain of Graph Theory. The conjecture proposes that every graph with at least three vertices can be uniquely reconstructed given the multiset of subgraphs produced by deleting each vertex of the original graph one by one. This conjecture has been proven true for several infinite classes of graphs, but the general case remains unsolved. In this talk, I will outline the problem, and introduce some of the most well known results. Graduate Seminar: Graph Reconstruction p. 2/2
3 Graph Theory The study of Graphs... Graduate Seminar: Graph Reconstruction p. 3/2
4 Graph Theory The study of Graphs... so what s a graph? Graduate Seminar: Graph Reconstruction p. 3/2
5 Graph Theory Graduate Seminar: Graph Reconstruction p. 3/2
6 Graph Theory V (G) = the set of vertices of G. Graduate Seminar: Graph Reconstruction p. 3/2
7 Graph Theory V (G) = the set of vertices of G. E(G) = the set of edges of G. Graduate Seminar: Graph Reconstruction p. 3/2
8 Definitions For v V (G), the degree of v, deg(v), is the number of edges connected to v. Graduate Seminar: Graph Reconstruction p. 4/2
9 Definitions For v V (G), the degree of v, deg(v), is the number of edges connected to v. A graph is regular if all it s vertices have the same degree. Graduate Seminar: Graph Reconstruction p. 4/2
10 Definitions For v V (G), the degree of v, deg(v), is the number of edges connected to v. A graph is regular if all it s vertices have the same degree. K n denotes the complete graph on n vertices (every possible edge). Graduate Seminar: Graph Reconstruction p. 4/2
11 Definitions For v V (G), the degree of v, deg(v), is the number of edges connected to v. A graph is regular if all it s vertices have the same degree. K n denotes the complete graph on n vertices (every possible edge). A card of G is an unlabelled graph formed by deleting 1 vertex and all edges attached to it. Graduate Seminar: Graph Reconstruction p. 4/2
12 Definitions For v V (G), the degree of v, deg(v), is the number of edges connected to v. A graph is regular if all it s vertices have the same degree. K n denotes the complete graph on n vertices (every possible edge). A card of G is an unlabelled graph formed by deleting 1 vertex and all edges attached to it. The deck of G, D(G), is the collection of all G s cards. Note this is in general a multiset. Graduate Seminar: Graph Reconstruction p. 4/2
13 Example A C B D Graduate Seminar: Graph Reconstruction p. 5/2
14 Example A C B D C B D Graduate Seminar: Graph Reconstruction p. 5/2
15 Example A C B D C A C B D D Graduate Seminar: Graph Reconstruction p. 5/2
16 Example A C B D C A C A B D D B D Graduate Seminar: Graph Reconstruction p. 5/2
17 Example A C B D C A C A A C B D D B D B Graduate Seminar: Graph Reconstruction p. 5/2
18 Example A C B D Graduate Seminar: Graph Reconstruction p. 5/2
19 Example A C B D Graduate Seminar: Graph Reconstruction p. 5/2
20 Example A C B D Graduate Seminar: Graph Reconstruction p. 5/2
21 Example A C B D A graph is reconstructible if, given it s deck, we can uniquely determine what the original graph was. Graduate Seminar: Graph Reconstruction p. 5/2
22 Example A C B D A graph is reconstructible if, given it s deck, we can uniquely determine what the original graph was. Equivalently, G is not reconstructible if and only if there is an H such that D(G) = D(H). Graduate Seminar: Graph Reconstruction p. 5/2
23 Example A C B D This graph is reconstructible. Graduate Seminar: Graph Reconstruction p. 6/2
24 Example A C B D This graph is reconstructible. The last three cards are also in the deck of Graduate Seminar: Graph Reconstruction p. 6/2
25 Example A C B D This graph is reconstructible. The last three cards are also in the deck of (these can be checked by exhaustion). Graduate Seminar: Graph Reconstruction p. 6/2
26 The Reconstruction Conjecture Reconstruction Conjecture: Every graph with at least 3 vertices is reconstructible. Graduate Seminar: Graph Reconstruction p. 7/2
27 The Reconstruction Conjecture Reconstruction Conjecture: Every graph with at least 3 vertices is reconstructible. We need at least three vertices: Graduate Seminar: Graph Reconstruction p. 7/2
28 The Reconstruction Conjecture Reconstruction Conjecture: Every graph with at least 3 vertices is reconstructible. We need at least three vertices: Note: the reconstruction conjecture is false if and only if there exist two graphs with the same deck. Graduate Seminar: Graph Reconstruction p. 7/2
29 Reconstructible Properties Some properties of G are easily determined from D(G): Graduate Seminar: Graph Reconstruction p. 8/2
30 Reconstructible Properties Some properties of G are easily determined from D(G): V (G) = # of cards in D(G). Graduate Seminar: Graph Reconstruction p. 8/2
31 Reconstructible Properties Some properties of G are easily determined from D(G): V (G) = # of cards in D(G). E(G) : Each edge in G is missing from exactly two cards. Thus each edge is in exactly n 2 cards, and we have E(G) = 1 n 2 v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 8/2
32 Further... Lemma: v V (G), deg(v) = E(G) E(G v). Graduate Seminar: Graph Reconstruction p. 9/2
33 Further... Lemma: v V (G), deg(v) = E(G) E(G v). Since E(G) can be determined from D(G), we can also determine the degree of v for all v V (G). Graduate Seminar: Graph Reconstruction p. 9/2
34 Further... Lemma: v V (G), deg(v) = E(G) E(G v). Since E(G) can be determined from D(G), we can also determine the degree of v for all v V (G). = We can determine the whole degree sequence of G. Graduate Seminar: Graph Reconstruction p. 9/2
35 Reconstruction Example Reconstruct, if possible, one or more graphs from the following deck: Graduate Seminar: Graph Reconstruction p. 10/2
36 Reconstruction Example V (G) = # of cards. Graduate Seminar: Graph Reconstruction p. 10/2
37 Reconstruction Example V (G) = 5. Graduate Seminar: Graph Reconstruction p. 10/2
38 Reconstruction Example V (G) = 5. Graduate Seminar: Graph Reconstruction p. 10/2
39 Reconstruction Example V (G) = 5. E(G) = 1 V (G) 2 v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 10/2
40 Reconstruction Example V (G) = 5. E(G) = 1 V (G) 2 v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 10/2
41 Reconstruction Example V (G) = 5. E(G) = v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 10/2
42 Reconstruction Example V (G) = 5. E(G) = 1 3 v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 10/2
43 Reconstruction Example V (G) = 5. E(G) = 1 3 v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 10/2
44 Reconstruction Example V (G) = 5. E(G) = 1 ( ). 3 Graduate Seminar: Graph Reconstruction p. 10/2
45 Reconstruction Example V (G) = 5. E(G) = 1 ( ) = 6. 3 Graduate Seminar: Graph Reconstruction p. 10/2
46 Reconstruction Example V (G) = 5. E(G) = 6. Graduate Seminar: Graph Reconstruction p. 10/2
47 Reconstruction Example V (G) = 5. E(G) = 6. Thus the degrees of the missing vertices in order are 1,4,2,2,3. Graduate Seminar: Graph Reconstruction p. 10/2
48 Reconstruction Example Note the degree of the vertex deleted in the second graph is 4. Graduate Seminar: Graph Reconstruction p. 11/2
49 Reconstruction Example Note the degree of the vertex deleted in the second graph is 4. Thus it must have been connected to every other vertex: Graduate Seminar: Graph Reconstruction p. 11/2
50 Reconstruction Example Note the degree of the vertex deleted in the second graph is 4. Thus it must have been connected to every other vertex: Graduate Seminar: Graph Reconstruction p. 11/2
51 Reconstruction Example Note the degree of the vertex deleted in the second graph is 4. Thus it must have been connected to every other vertex: Graduate Seminar: Graph Reconstruction p. 11/2
52 Reconstruction Example Note the degree of the vertex deleted in the second graph is 4. Thus it must have been connected to every other vertex: Determined Uniquely! Graduate Seminar: Graph Reconstruction p. 11/2
53 Recognizable Properties Let G have some property. This property is called recognizable if whenever we have a graph H with the same deck as G, H must also have the same property. Graduate Seminar: Graph Reconstruction p. 12/2
54 Recognizable Properties Let G have some property. This property is called recognizable if whenever we have a graph H with the same deck as G, H must also have the same property. Since we can determine the degree sequence uniquely given a deck, regular graphs are recognizable. Graduate Seminar: Graph Reconstruction p. 12/2
55 Recognizable Properties Let G have some property. This property is called recognizable if whenever we have a graph H with the same deck as G, H must also have the same property. Since we can determine the degree sequence uniquely given a deck, regular graphs are recognizable. A graph is regular if all it s vertices have the same degree. Graduate Seminar: Graph Reconstruction p. 12/2
56 Class of reconstructible graphs Theorem(Kelly, 1957) Regular graphs are reconstructible. Proof. Graduate Seminar: Graph Reconstruction p. 13/2
57 Class of reconstructible graphs Theorem(Kelly, 1957) Regular graphs are reconstructible. Proof. Let G be regular, and construct D(G). Graduate Seminar: Graph Reconstruction p. 13/2
58 Class of reconstructible graphs Theorem(Kelly, 1957) Regular graphs are reconstructible. Proof. Let G be regular, and construct D(G). By previous slide, the regularity of G is recognizable. Graduate Seminar: Graph Reconstruction p. 13/2
59 Class of reconstructible graphs Theorem(Kelly, 1957) Regular graphs are reconstructible. Proof. Let G be regular, and construct D(G). By previous slide, the regularity of G is recognizable. Let d be the degree of each vertex. Graduate Seminar: Graph Reconstruction p. 13/2
60 Class of reconstructible graphs Theorem(Kelly, 1957) Regular graphs are reconstructible. Proof. Let G be regular, and construct D(G). By previous slide, the regularity of G is recognizable. Let d be the degree of each vertex. Add a vertex to any card. Graduate Seminar: Graph Reconstruction p. 13/2
61 Class of reconstructible graphs Theorem(Kelly, 1957) Regular graphs are reconstructible. Proof. Let G be regular, and construct D(G). By previous slide, the regularity of G is recognizable. Let d be the degree of each vertex. Add a vertex to any card. Connect new vertex to each of the d vertices in the card with degree d 1, and G is reconstructed uniquely. Graduate Seminar: Graph Reconstruction p. 13/2
62 Kelly s Lemma s(f, G) = # of subgraphs of G isomorphic to F. Graduate Seminar: Graph Reconstruction p. 14/2
63 Kelly s Lemma s(f, G) = # of subgraphs of G isomorphic to F. Kelly s Theorem: For all F with less vertices than G, s(f, G) = 1 ( V (G) V (F) ) v V (G) s(f, G v). Graduate Seminar: Graph Reconstruction p. 14/2
64 Proof of Kelly s Theorem: Double count ordered pairs of the form (H, G v) where H = F, and H G v. Graduate Seminar: Graph Reconstruction p. 15/2
65 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... Graduate Seminar: Graph Reconstruction p. 16/2
66 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... E(G) = 1 V (G) V (K 2 ) v V (G) s(k 2, G v). Graduate Seminar: Graph Reconstruction p. 16/2
67 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... E(G) = 1 V (G) V (K 2 ) v V (G) s(k 2, G v). Graduate Seminar: Graph Reconstruction p. 16/2
68 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... E(G) = 1 V (G) 2 v V (G) s(k 2, G v). Graduate Seminar: Graph Reconstruction p. 16/2
69 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... E(G) = 1 V (G) 2 v V (G) s(k 2, G v). Graduate Seminar: Graph Reconstruction p. 16/2
70 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... E(G) = 1 V (G) 2 v V (G) E(G v). Graduate Seminar: Graph Reconstruction p. 16/2
71 A Nice Consequence of Kelly s Lemma Let F = K 2 (edges). Then s(k 2, G) = E(G), and... E(G) = 1 V (G) 2 v V (G) E(G v). Thus, we have that Kelly s Lemma implies that the number of edges is determined by the deck. Graduate Seminar: Graph Reconstruction p. 16/2
72 Even more However, Kelly s Lemma gives us even more: s(f, G) for any F. Graduate Seminar: Graph Reconstruction p. 17/2
73 Even more However, Kelly s Lemma gives us even more: s(f, G) for any F. Thus if there are two graphs with the same deck, they must have the same: Graduate Seminar: Graph Reconstruction p. 17/2
74 Even more However, Kelly s Lemma gives us even more: s(f, G) for any F. Thus if there are two graphs with the same deck, they must have the same: number of vertices. Graduate Seminar: Graph Reconstruction p. 17/2
75 Even more However, Kelly s Lemma gives us even more: s(f, G) for any F. Thus if there are two graphs with the same deck, they must have the same: number of vertices. number of edges. Graduate Seminar: Graph Reconstruction p. 17/2
76 Even more However, Kelly s Lemma gives us even more: s(f, G) for any F. Thus if there are two graphs with the same deck, they must have the same: number of vertices. number of edges. number of subgraphs of EVERY type (same s(f, G) for all F with V (F) < V (G) ). Graduate Seminar: Graph Reconstruction p. 17/2
77 Thoughts on the Conjecture So maybe one would think then that all graphs are reconstructible, and the conjecture is true? Graduate Seminar: Graph Reconstruction p. 18/2
78 Thoughts on the Conjecture So maybe one would think then that all graphs are reconstructible, and the conjecture is true? The conjecture has been confirmed after all for all graphs with less than 11 vertices ( 12.3 million graphs). Graduate Seminar: Graph Reconstruction p. 18/2
79 Thoughts on the Conjecture So maybe one would think then that all graphs are reconstructible, and the conjecture is true? The conjecture has been confirmed after all for all graphs with less than 11 vertices ( 12.3 million graphs). However, there are good reasons to believe it false... Graduate Seminar: Graph Reconstruction p. 18/2
80 False Reconstruction Conjectures Digraphs are not reconstructible in general. Graduate Seminar: Graph Reconstruction p. 19/2
81 False Reconstruction Conjectures Digraphs are not reconstructible in general. There are arbitrarily large counterexamples. (Stockmeyer, 1977/1981) Graduate Seminar: Graph Reconstruction p. 19/2
82 False Reconstruction Conjectures Digraphs are not reconstructible in general. There are arbitrarily large counterexamples. (Stockmeyer, 1977/1981) Hypergraphs are not, in general, reconstructible. Graduate Seminar: Graph Reconstruction p. 19/2
83 False Reconstruction Conjectures Digraphs are not reconstructible in general. There are arbitrarily large counterexamples. (Stockmeyer, 1977/1981) Hypergraphs are not, in general, reconstructible. There is an infinite family of counterexamples. (Kocay, 1987) Graduate Seminar: Graph Reconstruction p. 19/2
84 False Reconstruction Conjectures Digraphs are not reconstructible in general. There are arbitrarily large counterexamples. (Stockmeyer, 1977/1981) Hypergraphs are not, in general, reconstructible. There is an infinite family of counterexamples. (Kocay, 1987) Infinite graphs are not, in general, reconstructible. Graduate Seminar: Graph Reconstruction p. 19/2
85 Reconstruction Numbers Related Topic: Graph Reconstruction Numbers. Graduate Seminar: Graph Reconstruction p. 20/2
86 Reconstruction Numbers Related Topic: Graph Reconstruction Numbers. The existential reconstruction number of G, rn(g), is the minimum number of cards in D(G) required to reconstruct G uniquely. Graduate Seminar: Graph Reconstruction p. 20/2
87 Reconstruction Numbers Related Topic: Graph Reconstruction Numbers. The existential reconstruction number of G, rn(g), is the minimum number of cards in D(G) required to reconstruct G uniquely. The universal reconstruction number of G, rn(g), is the minimum number n such that all multisubsets of D(G) of size n uniquely reconstruct G. Graduate Seminar: Graph Reconstruction p. 20/2
88 Reconstruction Numbers Related Topic: Graph Reconstruction Numbers. The existential reconstruction number of G, rn(g), is the minimum number of cards in D(G) required to reconstruct G uniquely. The universal reconstruction number of G, rn(g), is the minimum number n such that all multisubsets of D(G) of size n uniquely reconstruct G. Almost all graphs have rn(g) = 3. (Bollobas, 1990). Graduate Seminar: Graph Reconstruction p. 20/2
89 Example A B C D The last three cards in this deck are as well in the deck of. Graduate Seminar: Graph Reconstruction p. 21/2
90 Example A B C D The last three cards in this deck are as well in the deck of However, the first three graphs DO determine the graph G uniquely.. Graduate Seminar: Graph Reconstruction p. 21/2
91 Example A B C D The last three cards in this deck are as well in the deck of However, the first three graphs DO determine the graph G uniquely. Thus, for this G, rn(g) = 3, rn(g) = 4.. Graduate Seminar: Graph Reconstruction p. 21/2
92 Reconstruction Numbers Found for all graphs on at most 10 vertices: rn n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = , ,666 12,005, Graduate Seminar: Graph Reconstruction p. 22/2
93 Reconstruction Numbers Found for all graphs on at most 10 vertices: rn n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = , ,666 12,005, Very hard to get any further: there are 1,018,997,864 graphs on 11 vertices! Graduate Seminar: Graph Reconstruction p. 22/2
94 Other Related Problems The Legitimate Deck Problem: Given a multiset of graphs G, does there exist a graph H such that D(H) = G? Graduate Seminar: Graph Reconstruction p. 23/2
95 Other Related Problems The Legitimate Deck Problem: Given a multiset of graphs G, does there exist a graph H such that D(H) = G? The problem of k-reconstruction: Instead of deleting 1 vertex to form each card, delete k vertices. Then determine f(k), if it exists, such that all graphs with greater than f(k) vertices are k-reconstructible. Graduate Seminar: Graph Reconstruction p. 23/2
96 Other Related Problems The Legitimate Deck Problem: Given a multiset of graphs G, does there exist a graph H such that D(H) = G? The problem of k-reconstruction: Instead of deleting 1 vertex to form each card, delete k vertices. Then determine f(k), if it exists, such that all graphs with greater than f(k) vertices are k-reconstructible. The Graph Reconstruction Conjecture says f(1) = 2. Graduate Seminar: Graph Reconstruction p. 23/2
97 References Bollobás, B., Almost every graph has reconstruction number three. J. Graph Theory 14 (1990), 1 4. Bondy, J. A., A Graph Reconstructor s Manual, in Survey s in Combinatorics, Cambridge University Press (1991). Kocay, W. L., A Family of Nonreconstructible Hypergraphs, J. Comb. Thy (B) 42 (1987), McMullen, B., Graph Reconstruction Numbers, Master s Project Report, Rochester Institute of Technology (2005). Graduate Seminar: Graph Reconstruction p. 24/2
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